Answer :
To find out how many hours it will take for the number of bacteria to reach 3700, we need to solve for [tex]\( h \)[/tex] in the function [tex]\( P(h) = 2800 e^{0.09h} \)[/tex].
Here are the steps:
1. Set up the equation:
We are given the formula for the bacteria population:
[tex]\[
P(h) = 2800 e^{0.09h}
\][/tex]
We want to find [tex]\( h \)[/tex] when [tex]\( P(h) = 3700 \)[/tex]. Thus, set the equation:
[tex]\[
2800 e^{0.09h} = 3700
\][/tex]
2. Divide both sides by 2800:
Doing this isolates the exponential part of the equation:
[tex]\[
e^{0.09h} = \frac{3700}{2800}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{3700}{2800} = \frac{37}{28}
\][/tex]
4. Take the natural logarithm of both sides:
Using the property of logarithms, we can bring down the exponent:
[tex]\[
\ln(e^{0.09h}) = \ln\left(\frac{37}{28}\right)
\][/tex]
5. Simplify the left side:
The natural log of [tex]\( e \)[/tex] to a power simplifies to that power:
[tex]\[
0.09h = \ln\left(\frac{37}{28}\right)
\][/tex]
6. Solve for [tex]\( h \)[/tex]:
Divide both sides by 0.09 to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{\ln\left(\frac{37}{28}\right)}{0.09}
\][/tex]
7. Calculate the result:
First, find the natural logarithm:
[tex]\[
\ln\left(\frac{37}{28}\right) \approx 0.290499
\][/tex]
Then divide by 0.09:
[tex]\[
h \approx \frac{0.290499}{0.09} \approx 3.22777
\][/tex]
8. Round to the nearest tenth:
The result [tex]\( h \approx 3.23 \)[/tex] rounds to [tex]\( 3.2 \)[/tex].
So, it will take approximately 3.2 hours for the bacteria population to reach 3700.
Here are the steps:
1. Set up the equation:
We are given the formula for the bacteria population:
[tex]\[
P(h) = 2800 e^{0.09h}
\][/tex]
We want to find [tex]\( h \)[/tex] when [tex]\( P(h) = 3700 \)[/tex]. Thus, set the equation:
[tex]\[
2800 e^{0.09h} = 3700
\][/tex]
2. Divide both sides by 2800:
Doing this isolates the exponential part of the equation:
[tex]\[
e^{0.09h} = \frac{3700}{2800}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{3700}{2800} = \frac{37}{28}
\][/tex]
4. Take the natural logarithm of both sides:
Using the property of logarithms, we can bring down the exponent:
[tex]\[
\ln(e^{0.09h}) = \ln\left(\frac{37}{28}\right)
\][/tex]
5. Simplify the left side:
The natural log of [tex]\( e \)[/tex] to a power simplifies to that power:
[tex]\[
0.09h = \ln\left(\frac{37}{28}\right)
\][/tex]
6. Solve for [tex]\( h \)[/tex]:
Divide both sides by 0.09 to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{\ln\left(\frac{37}{28}\right)}{0.09}
\][/tex]
7. Calculate the result:
First, find the natural logarithm:
[tex]\[
\ln\left(\frac{37}{28}\right) \approx 0.290499
\][/tex]
Then divide by 0.09:
[tex]\[
h \approx \frac{0.290499}{0.09} \approx 3.22777
\][/tex]
8. Round to the nearest tenth:
The result [tex]\( h \approx 3.23 \)[/tex] rounds to [tex]\( 3.2 \)[/tex].
So, it will take approximately 3.2 hours for the bacteria population to reach 3700.
